Scaling functions generating fractional Hilbert transforms of a wavelet function

Abstract

It is well-known that an orthonormal scaling function generates an orthonormal wavelet function in the theory of multiresolution analysis. We consider two families of unitary operators. One is a family of extensions of the Hilbert transform called fractional Hilbert transforms. The other is a new family of operators which are a kind of modified translation operators. A fractional Hilbert transform of a given orthonormal wavelet (resp. scaling) function is also an orthonormal wavelet (resp. scaling) function, although a fractional Hilbert transform of a scaling function has bad localization in many cases. We show that a modified translation of a scaling function is also a scaling function, and it generates a fractional Hilbert transform of the corresponding wavelet function. We also show a good localization property of the modified translation operators. The modified translation operators act on the Meyer scaling functions as the ordinary translation operators. We give a class of scaling functions, on which the modified translation operators act as the ordinary translation operators.